\(\int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx\) [393]

Optimal result

Integrand size = 21, antiderivative size = 248 \[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {5 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{4 f}-\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}{1+\sqrt {2}+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f}-\frac {5 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f} \] Output:

(1+2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)-2*(1+tan(f*x+e))^(1/2))/(-2+ 
2*2^(1/2))^(1/2))/f-(1+2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)+2*(1+tan 
(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))/f+5/4*arctanh((1+tan(f*x+e))^(1/2))/ 
f-arctanh((2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)/(1+2^(1/2)+tan(f*x+e))) 
/(1+2^(1/2))^(1/2)/f-5/4*cot(f*x+e)*(1+tan(f*x+e))^(1/2)/f-1/2*cot(f*x+e)^ 
2*(1+tan(f*x+e))^(1/2)/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.50 \[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {-5 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+4 (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+4 (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+5 \cot (e+f x) \sqrt {1+\tan (e+f x)}+2 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \] Input:

Integrate[Cot[e + f*x]^3*(1 + Tan[e + f*x])^(3/2),x]
 

Output:

-1/4*(-5*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 4*(1 - I)^(3/2)*ArcTanh[Sqrt[1 
+ Tan[e + f*x]]/Sqrt[1 - I]] + 4*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f* 
x]]/Sqrt[1 + I]] + 5*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 2*Cot[e + f*x]^ 
2*Sqrt[1 + Tan[e + f*x]])/f
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.36, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4050, 27, 3042, 4133, 27, 3042, 4136, 27, 3042, 3966, 484, 1407, 1142, 25, 1083, 217, 1103, 4117, 73, 220}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (\tan (e+f x)+1)^{3/2} \cot ^3(e+f x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(\tan (e+f x)+1)^{3/2}}{\tan (e+f x)^3}dx\)

\(\Big \downarrow \) 4050

\(\displaystyle -\frac {1}{2} \int -\frac {\cot ^2(e+f x) \left (5-3 \tan ^2(e+f x)\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {\cot ^2(e+f x) \left (5-3 \tan ^2(e+f x)\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \int \frac {5-3 \tan (e+f x)^2}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4133

\(\displaystyle \frac {1}{4} \left (-\int \frac {\cot (e+f x) \left (5 \tan ^2(e+f x)+16 \tan (e+f x)+5\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {\cot (e+f x) \left (5 \tan ^2(e+f x)+16 \tan (e+f x)+5\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {5 \tan (e+f x)^2+16 \tan (e+f x)+5}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\int \frac {16}{\sqrt {\tan (e+f x)+1}}dx-5 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-16 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx-5 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-16 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3966

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {16 \int \frac {1}{\sqrt {\tan (e+f x)+1} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{f}-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 484

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \int \frac {1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 1407

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\int \frac {\sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\frac {1}{2} \int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {32 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {5 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {10 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {10 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {5 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

Input:

Int[Cot[e + f*x]^3*(1 + Tan[e + f*x])^(3/2),x]
 

Output:

-1/2*(Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/f + (((10*ArcTanh[Sqrt[1 + Ta 
n[e + f*x]]])/f - (32*((Sqrt[(1 + Sqrt[2])/(-1 + Sqrt[2])]*ArcTan[(-Sqrt[2 
*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]] - Log[ 
1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]] 
/2)/(4*Sqrt[1 + Sqrt[2]]) + (Sqrt[(1 + Sqrt[2])/(-1 + Sqrt[2])]*ArcTan[(Sq 
rt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]] + 
Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f* 
x]]]/2)/(4*Sqrt[1 + Sqrt[2]])))/f)/2 - (5*Cot[e + f*x]*Sqrt[1 + Tan[e + f* 
x]])/f)/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 220

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 
1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && 
 (LtQ[a, 0] || GtQ[b, 0])
 

rule 484

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* 
d   Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], 
 x] /; FreeQ[{a, b, c, d}, x]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1407

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ 
c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   Int[(r - x)/(q - r* 
x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(r + x)/(q + r*x + x^2), x], x]]] 
 /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3966

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Su 
bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c 
, d, n}, x] && NeQ[a^2 + b^2, 0]
 

rule 4050

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 
 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m 
 + 1)*(a^2 + b^2))   Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ 
(n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 
*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2 
, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^ 
2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[ 
2*m]
 

rule 4117

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
 Simp[A/f   Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; 
FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
 

rule 4133

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n 
+ 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d) 
*(a^2 + b^2))   Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Sim 
p[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n 
 + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*( 
m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, 
 x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m 
, -1] &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

rule 4136

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) 
+ (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2)   Int[(c + d*Tan[e + f*x])^ 
n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ 
(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)   Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ 
e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 
 C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & 
&  !GtQ[n, 0] &&  !LeQ[n, -1]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(196)=392\).

Time = 170.88 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.60

method	result	size
default	\(\text {Expression too large to display}\)	\(894\)

Input:

int(cot(f*x+e)^3*(tan(f*x+e)+1)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8/f*(tan(f*x+e)+1)^(1/2)/((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(1+cos(f*x 
+e))^2)^(1/2)/(1+cos(f*x+e))/(cot(f*x+e)+cot(f*x+e)^2)^(1/2)*(4*(-2+2*2^(1 
/2))^(1/2)*(1+2^(1/2))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x 
+e))*cos(f*x+e)*(3*2^(1/2)-4)/(2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin(f*x+e 
)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)/(cos(f*x+e)+s 
in(f*x+e))*(2*cos(f*x+e)-2*sin(f*x+e)-sec(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2* 
2^(1/2)+3)*(3*2^(1/2)-4))*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*s 
in(f*x+e)*cos(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin( 
f*x+e)^2+1))^(1/2)*(2^(1/2)*(cot(f*x+e)*cos(f*x+e)-3*cos(f*x+e))-2*cot(f*x 
+e)*cos(f*x+e)+4*cos(f*x+e))+5*cos(f*x+e)*(1+2^(1/2))^(1/2)*(-2*2^(1/2)+3) 
*(cot(f*x+e)+cot(f*x+e)^2)^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))* 
cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*c 
os(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-2*cot(f*x+e)-1)+8*((cos(f*x+e)+sin(f*x+e 
))*cos(f*x+e)/(2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*si 
n(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+ 
e))*cos(f*x+e)/(2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*s 
in(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*( 
2^(1/2)*(-2*cot(f*x+e)*cos(f*x+e)+5*cos(f*x+e))+3*cot(f*x+e)*cos(f*x+e)-7* 
cos(f*x+e))+2*cot(f*x+e)*csc(f*x+e)*(1+2^(1/2))^(1/2)*(2*(5*(-cos(f*x+e)-1 
)*sin(f*x+e)-2*cos(f*x+e)*(1+cos(f*x+e)))*2^(1/2)+3*(1+cos(f*x+e))*(5*s...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (196) = 392\).

Time = 0.12 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.71 \[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {4 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} + 4 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 5 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{2} + 5 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{2} + 2 \, {\left (5 \, \tan \left (f x + e\right ) + 2\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{8 \, f \tan \left (f x + e\right )^{2}} \] Input:

integrate(cot(f*x+e)^3*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

-1/8*(4*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(2)*(f^3*sqrt( 
-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 1)) 
*tan(f*x + e)^2 - 4*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(-sqrt( 
2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f 
*x + e) + 1))*tan(f*x + e)^2 - 4*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2 
)*log(sqrt(2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2) + 2* 
sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 + 4*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4 
) - 1)/f^2)*log(-sqrt(2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1 
)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 - 5*log(sqrt(tan(f*x + e 
) + 1) + 1)*tan(f*x + e)^2 + 5*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e 
)^2 + 2*(5*tan(f*x + e) + 2)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^2)
 

Sympy [F]

\[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot ^{3}{\left (e + f x \right )}\, dx \] Input:

integrate(cot(f*x+e)**3*(1+tan(f*x+e))**(3/2),x)
 

Output:

Integral((tan(e + f*x) + 1)**(3/2)*cot(e + f*x)**3, x)
 

Maxima [F]

\[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int { {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{3} \,d x } \] Input:

integrate(cot(f*x+e)^3*(1+tan(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate((tan(f*x + e) + 1)^(3/2)*cot(f*x + e)^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(f*x+e)^3*(1+tan(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.57 \[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {\frac {3\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{4}-\frac {5\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{4}}{f-2\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2}-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{4\,f}-\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \] Input:

int(cot(e + f*x)^3*(tan(e + f*x) + 1)^(3/2),x)
 

Output:

((3*(tan(e + f*x) + 1)^(1/2))/4 - (5*(tan(e + f*x) + 1)^(3/2))/4)/(f - 2*f 
*(tan(e + f*x) + 1) + f*(tan(e + f*x) + 1)^2) - (atan((tan(e + f*x) + 1)^( 
1/2)*1i)*5i)/(4*f) - atan(f*((- 1/2 - 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^ 
(1/2))*((- 1/2 - 1i/2)/f^2)^(1/2)*2i + atan(f*((- 1/2 + 1i/2)/f^2)^(1/2)*( 
tan(e + f*x) + 1)^(1/2))*((- 1/2 + 1i/2)/f^2)^(1/2)*2i
 

Reduce [F]

\[ \int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \sqrt {\tan \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{3} \tan \left (f x +e \right )d x +\int \sqrt {\tan \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{3}d x \] Input:

int(cot(f*x+e)^3*(1+tan(f*x+e))^(3/2),x)
 

Output:

int(sqrt(tan(e + f*x) + 1)*cot(e + f*x)**3*tan(e + f*x),x) + int(sqrt(tan( 
e + f*x) + 1)*cot(e + f*x)**3,x)